Hierarchical Models, Marginal Polytopes, and Linear Codes

In this paper, we explore a connection between binary hierarchical models, convex geometry, and coding theory. Using the so called moment map, each hierarchical model is mapped to a convex polytope, the marginal polytope. We realize the marginal polytopes as 0/1-polytopes and show that their vertices form a linear code. We determine a class of linear codes that is realizable by hierarchical models. The realization inside the unit cube also allows for a connection to other classes of 0/1-polytopes. We show that the hierarchical models of pair interactions give exactly the CUT-polytopes of complete graphs, and generalize their affine equivalence to correlation polytopes. Finally, we classify all full dimensional polytopes with the property that their vertices form a linear code and give an algorithm that determines them.